\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[all,cmtip]{xy}
\usepackage[english]{babel}
\usepackage{graphicx}

\begin{document}

\author{Conor Mahany}
\title{Problem Set 2, ECON 24210/40501}
\maketitle

1. a. Note that each term in the sum $-(x_U - 2 x_C)^2 - x_C^4 -(x_C - 2 x_U)^2 - x_U^4$ is less than or equal to 0, so the sum is maximized when $-x_C^4 = -x_U^4 = 0,$ that is, when $x_C = x_U = 0$, or both countries are perfectly indifferent between the two.

\vspace{10pt}

b. China faces the optimization problem $\max_{x_C} \{ -(x_C - 2 x_U)^2 - x_U^4 \}$. Setting the derivative with respect to $x_C$ equal to 0 yields the root
\[
x_C^*(x_U) = \frac{ \sqrt[3]{ \sqrt{6} \sqrt{ 54 x_U^2 + 1} + 18x_U}}{6^{2/3}} - \frac{1}{\sqrt[3]{6} \sqrt[3]{ \sqrt{6} \sqrt{54 x_U^2 + 1} + 18 x_U^2}}
\]
and by symmetry
\[
x_U^*(x_C) = \frac{ \sqrt[3]{ \sqrt{6} \sqrt{ 54 x_C^2 + 1} + 18x_C}}{6^{2/3}} - \frac{1}{\sqrt[3]{6} \sqrt[3]{ \sqrt{6} \sqrt{54 x_C^2 + 1} + 18 x_C^2}}
\]


\end{document}